Question: Simplify $\displaystyle\frac{2+2i}{-3+4i}$.  Express your answer as a complex number in the form $a+bi$, where $a$ and $b$ are real numbers.
Answer: Multiplying the numerator and denominator by the conjugate of the denominator, we have \begin{align*}
\frac{2+2i}{-3+4i} \cdot \frac{-3-4i}{-3-4i} &= \frac{2(-3) + 2(-4i) - 3(2i) + 2i(-4i)}{-3(-3) -3(4i) + 3(4i) -4i(4i)} \\
&= \frac{2-14i}{25} \\
&= \boxed{\frac{2}{25} - \frac{14}{25}i}.
\end{align*}